Problem: $ A = \left[\begin{array}{rr}1 & 0 \\ -1 & 3\end{array}\right]$ $ v = \left[\begin{array}{r}4 \\ 1\end{array}\right]$ What is $ A v$ ?
Because $ A$ has dimensions $(2\times2)$ and $ v$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(2\times1)$ $ A v = \left[\begin{array}{rr}{1} & {0} \\ {-1} & {3}\end{array}\right] \left[\begin{array}{r}{4} \\ {1}\end{array}\right] = \left[\begin{array}{r}? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ v$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ v$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ v$ , and so on. Add the products together. $ \left[\begin{array}{r}{1}\cdot{4}+{0}\cdot{1} \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ v$ and add the products together. $ \left[\begin{array}{r}{1}\cdot{4}+{0}\cdot{1} \\ {-1}\cdot{4}+{3}\cdot{1}\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{1}\cdot{4}+{0}\cdot{1} \\ {-1}\cdot{4}+{3}\cdot{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}4 \\ -1\end{array}\right] $